There's an old example I've talked about elsewhere, a thought experiment I remember first hearing from Feynman, but it shows up in Modern Physics books as a simple problem as well, is one where you have two identical particles moving at equal speeds towards one another in your reference frame, eventually colliding in a totally inelastic collision so that the resultant particle is stationary, you find that resultant conglomeration is heavier than the sum of the rest masses of the individual particles (but exactly equal to the sum of their relativistic masses).

This is because their energy is real, their total energy has to remain the same, and as a stationary system, the particles only have their rest mass energy remaining. Elementary.

People do have a problem with this because any real particles (as opposed to the point particles we play with in physics) going that fast will have a very nasty collision will all sorts of crazy interactions and particles flying off from its center. But that's not really important: if that bothers you just imagine you count up the total energy of all the craziness, it will the the same number as above.

Now a question I like to ask is what if these particles had accelerated from rest through an attractive force? What is the total energy of the initial state? I'm pretty sure of the answer, but I don't like it because it depends on the force.

If your particles were equally and oppositely charged and attracted by the electromagnetism, the electromagnetic field between those two objects should weigh just as much as they will increase in mass as they accelerate toward one another. (The field energy is basically the sum of the squares of the electric and magnetic fields.)

Fair enough. Makes sense.

But, some of our recent episodes have got me worried. We've done several dealing with equivalence principles, which got me to thinking. But one of the papers we read about the positive energy theorem acknowledged the problem head on (although in a different context):

If your particles are accelerated gravitationally, the Strong Equivalence Principle (and GR obeys SEP; it may be the only viable theory of gravity that does) states that the gravitationally attraction can't affect itself -- kindly avoiding all sorts of problems with highly nonlinear systems. That is, the gravitational field doesn't weigh anything.

So, the electromagnetically attracted system keeps the same mass throughout the process, and the gravitationally attracted pair changes its mass during the process.

If I understand your description properly, then you’re comparing the electric field (which has its own potential energy) with the gravitational field (which you’re saying doesn’t have its own energy).

But as I see it, the gravitational field energy is a feature of the inertial mass of the particle, so it’s there, but sort of in disguise. So the total energy (potential plus kinetic) is constant in both cases. The inertial mass of both particles decreases as their kinetic energy increases, to the same degree. So if the particles collide inelastically, then their increased gravitational binding energy (mass reduction) is perfectly balanced by their increased thermal energy post-collision, resulting in no net change.

Does that make sense, or am I missing something? Since both electrical fields and gravitational fields are conservative, then there can’t be any change in total energy, right?